Polynomial Chaos-based analysis of Multirotor Aerial Vehicles ModelSubjected to Uncertainties

Nome do primeiro autor, e-mail1Nome do segundo autor, e-mail1Nome do terceiro autor, e-mail2

1Nome da instituição, endereço para correspondência2Nome da instituição, endereço para correspondênciaCON-2016-1213

Abstract: Uncertainties are ubiquitous in mathematical equations that represent physical models and may come fromunknown plant parameters or from the purposeful choice of a simplified representation of the system dynamics. In caseof Multirotor Aerial Vehicles (MAVs), which have become interesting for applications where manned operations areconsidered inefficient and dangerous for humans, uncertainties such as inaccurate parameters, neglect of gyroscopiceffect, blade flapping, as well as, wind gusts can have strong adverse effects on the system behavior. This paper uses thePolynomial Chaos Expansion method to propagate uncertainties in the multirotor model and characterize the effects overthe vehicle trajectory, considering constraints on both the total thrust magnitude and the inclination of the rotor planein the design of the position control. The Polynomial Chaos method constructs meta-models that accurately mimic thebehavior of systems with uncertainty about the mean of stochastic inputs. The uncertainties are described in terms ofnormal distribution. The results are compared with Monte Carlo simulations.Keywords: Uncertainty, Polynomial Chaos, Multirotor Aerial Vehicles

1. INTRODUCTION

Among the different types of UAVs such as blimps (Elfes et al., 1998), fixed-wings (Beard et al., 2005) and rotarywings aircrafts (Bouabdallah et al., 2004), we highlight the multirotors (Gupte et al., 2012; Er et al., 2013). The multirotor-type unmanned aerial vehicle (MAVs) technology has attracted a great deal of interest of the academia and industry and,consequently experienced a rapid improvement. This interest is justified by features such as their simplified mechanics,low cost, high maneuverability, and vertical take-off and landing (VTOL) capability. Examples of their applicationsare found in building exploration (Achtelika et al., 2008), forest-fire monitoring (Alexis et al., 2012) and mapping ofagricultural areas (Barrientos et al., 2011).

Although there is a massive amount of concluded and ongoing research works on multirotor helicopters, the designof control laws for such vehicles still has challenges to be overcome. Recently, more efforts have been directed towardthe research of dealing with uncertainties associated with MAVs dynamics (Ton and MacKunis, 2012; Zhao et al., 2014).However, a good robust control design requires a deep understanding about the effects caused by the presence of uncer-tainties

Uncertainty quantification (UQ) is the characterization of the effects of uncertainties on simulation or theoretical mo-dels of actual systems. Sources of uncertainty include parametric model perturbations, lack of physical fidelity of models,and uncertain circumstances in system operation (Sandoval et al., 2012). The UQ can provide important information andperspectives for robust design and optimization and can be used to characterize the robustness, reachability and control-lability of the system. Furthermore, in the presence of multiple sources of uncertainty, efficient robust control requiresanalyses of their relative impacts on certain system performance and behavior (Kim et al., 2013).

Much research effort has been devoted to developing UQ methods. One of the well-known dynamic sampling methodsis Monte Carlo simulation. This approach typically involves the analysis of a large number of simulations runs of ananalytical or numerical system model with various combinations of parameters (Kewlani et al., 2012). However, toensure representation of the entire parameter range, a large number of simulations must be performed, often resultingin extensive computational costs. More recent approaches to stochastic simulation of processes subject to uncertaintyinclude Polynomial Chaos Expansion (PCE) approach, which was originally introduced by Wienner (1938). PCEs have

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become broadly used in a wide variety of fields including in stochastic differential equations (Xiu and Karniadakis, 2002),fluid dynamics (Walters, 2003), aircraft operations (Roberts et al., 2011), robust design problems (Dodson and Parks,2009). This approach can be considered as a spectral method to construct finite-dimensional approximations in infinite-dimensional probability measure space.

When constraints must be considered, model predictive control (MPC) algorithms appear as an interesting choice.The MPC is a strategy wherein a future control sequence is obtained by minimizing a cost function based on predictionsof the controlled output along a finite horizon typically, subject to constraints (Camacho and Bordons, 1998). Recently,Prado and Santos (2014) tackled the problem of controlling the position of a multirotor while respecting constraints onthe inclination of the rotor plane and on the magnitude of the total thrust vector. The present paper proposes to use thePCE to make the uncertainty quantification of a MAV position control designed by Prado and Santos (2014) consideringthe presence of wind gust with normal distribution. The disturbance effects are propagated to the outputs.

2. MAV MATHEMATICAL MODEL

This section presents the equations of motion of a multirotor aerial vehicle (MAV), considering all the relevant effectsnecessary to be accounted for in ground-thruth model for simulation-based evaluation of control laws. We start withpreliminary definitions in Subsection 2.1, then we present the rotor dynamics in 2.2 and the MAV rotational dynamics in2.3.

2.1 Preliminary definitions

We define two Cartesian coordinate systems (CCS) as illustrated in Fig. 1. The body CCS, SB , {XB, YB, ZB} isattached to the vehicle’s body with the origin at the vehicle’s center of mass (CM), the XB axis pointing forward, the ZBaxis pointing upward, perpendicular to the plane of the rotors, and the YB axis completing a dextrogeous frame. The CCS,SG = {XG, YG, ZG} is fixed to the ground at a known point O, with the ZG axis pointing upward, aligned with the localvertical.

Local Vertical

XB

ZB

f3

f2

f1

f4

CM

f

r

O

𝜙

ZR'

XR'

YR'

Local Horizontal

ZG

XG

YG

YB

Figura 1: The Cartesian coordinate systems (CCS). SB = {XB, YB, ZB} is the body CCS and SG = {XG, YG, ZG} isthe ground CCS.

2.2 Rotor dynamics

The thrust force fi and reaction torque τi produced by each individual rotor are modeled, respectively, by the aerody-namic models (Mahony et al., 2012)

fi = kfω2i , (1)

τi = kτω2i , (2)

for i = 1, ..., 4, where kf is the force coefficient, kτ is the torque coefficient, and ωi is the rotation speed of the ith rotor(positive in courterclockwise direction), whose dynamics can be modeled by the following first-order linear model:

ω̇i = −1

τωωi +

kωτωω̄i (3)

where ω̄i ∈ [0, ω̄max] is the rotation speed command for the ith rotor, kω is the rotation coefficient, and τω is rotor timeconstant.

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2.3 Resultant control thrust and torque

Although the formulation given here could be generalized to any multirotor vehicle configuration, for illustrationpurpose, we consider a quadrotor configuration with the longitudinal axis XB pointing between rotor 1 and rotor 2, andwith 45 degree of separation angle between adjacent arms; we call it the "x"configuration. Moreover, we consider thatrotor 1 and rotor 3 rotate in clockwise direction, while rotor 2 and rotor 4 rotate in counterclockwise direction. For theaforementioned configuration, one can show that the relationship between the individual rotor thrusts fi and the resultantthrust magnitude and torque vector is given by[

F c

Tc]

= Γf, (4)

with f ,[f1 f2 f3 f4

]Tand

Γ ,

1 1 1 1

l/√

2 −l/√

2 −l/√

2 l/√

2

−l/√

2 −l/√

2 l/√

2 l/√

2kτ/kf −kτ/kf kτ/kf −kτ/kf

, (5)where l is the length of each vehicle’s arm. The superscript c in Eq. (5) stands for the control command defined by theoperator and is used to distinguish from the other sources of force and torque that will be presented soon.

By inverting Eq. (5), one can compute the command f̄ ,[f̄1 f̄2 f̄3 f̄4

]Tas

f̄ = Ξ[F̄ c

T̄c], (6)

where Ξ , Γ−1, F̄ c is the thrust magnitude command, and T̄c is the control torque command.

2.4 Rotational motion

Representing the attitude using the Euler angles α ,[φ θ ψ

]Tin the rotation sequence 1-2-3, we have the

following rotational kinematics equations:

α̇ = AΩ (7)

where Ω , [Ωx Ωy Ωz]T is the SB representation of the vehicle’s angular velocity w.r.t. SG and

A ,

cosψ/ cos θ − sinψ/ cos θ 0sinψ cosψ 0− cosψ sin θ/ cos θ sinψ sin θ/ cos θ 1

(8)Assume that the vechile has a rigid structure and SG is an inertial frame. Considering the existence of gyroscopic

effects, due to the rotors, and disturbance torques, and using the Newton-Euler formulation, one can model the rotationaldynamics of the MAV by

Ω̇ = J−1(JΩ)×Ω + IrJ−1Ω× e34∑i=1

(−1)iωi + J−1Tc + J−1Td (9)

where Tc and Td are the SB representations of the resultant control torque and disturbance torque, respectively, e3 ,[0 0 1]T, Ir is the moment of inertia of the rotors w.r.t. the rotation axis, and J is the body inertia matrix. Consider thatthe vehicle has a symmetric structure with known mass m and inertia matrix in SB

J =

Jx 0 00 Jy 00 0 Jz

. (10)This paper makes use of the AR Drone 2 parameters in order to emulate a real plataform. The values can be seen in

Table 1.

2.5 Translational motion

Consider that the Earth is flat and the gravitational acceleration g is constant anywhere the vehicle is supposed tooperate. Therefore, the SG representation of the gravitational acceleration vector is given by

g =

00−g

. (11)

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Tabela 1: Parameters of AR Drone 2.Variables Values Unit

Mass of AR Drone 2, m 0.429 kgAcceleration of gravity, g 9.80665 m/s2

Arm lenght, l 0.1785 mMoment of inertia for x-axis, Jx 2.238× 10−3 kg.m2Moment of inertia for y-axis, Jy 2.986× 10−3 kg.m2Moment of inertia for z-axis, Jz 4.804× 10−3 kg.m2

Moment of inertia of each rotor, Ir 2.030× 10−5 kg.m2Force coefficient kf 8.050× 10−6 N/(rad/s)2

Torque coefficient, kτ 2.423× 10−7 N.m/(rad/s)2Time constant, τω 4.718× 10−3 s

Maximum speed of the motor, ωmax 1047 rad/s

Denote the SG representations of the position and velocity of SB w.r.t. SG by r = [rx ry rz]T and v = [vx vy vz]T,respectively. Therefore we can describe the vehicle’s translational kinematics and dynamics by

ṙ = v, (12)

v̇ =F c

mn + g +

1

mFd, (13)

where n , [ sinφ − sinφ cos θ cosφ cos θ ]T is the SG representation of the unit vector perpendicular to the planeof the rotors, Fd is the SG representation of the disturbance force, m is the vehicle’s total mass, and F c is the magnitudeof the resultant control thrust.

3. MODEL PREDICTIVE CONTROLLER

The present work faces the problem of safely controlling the position trajectory of multirotor UAVs by taking intoconsideration a conic constraint on the total thrust vector (see Fig 2a) and considering the presence of disturbance forcesFd. The problem is solved using a linear state-space model predictive control (MPC) strategy, whose optimization ismade handy by replacing the original conic constraint set on the thrust vector by an inscribed pyramidal space (seeFig 2b), which renders a linear set of inequalities. The control vector computed by the MPC is converted into a thrustmagnitude command and an attitude command. For more details about control strategy, consult the reference Prado andSantos (2014).

𝑍R′

𝑍B

𝜙max

𝐟

𝜙

CM

𝑓max

𝑓min

(a)

𝒁𝐁

𝒁𝐑′

𝐟

𝜙

𝜙max

𝑓max cos𝜙max

CM

𝑓z

intermediate

perimeter

𝑓min

(b)

Figura 2: (a) Original constraints space; (b) Linearized constraints space.

The constraints on both magnitude and inclination of f are directly connected to the vertical and lateral accelerationsof the vehicle. As one can see in Fig. 3, the component fz is responsible for controlling the altitude of the vehicle, whilefxy produces the lateral acceleration that leads the vehicle along the XR and YR directions, where fxy , [fx fy]T ∈ R2denotes the horizontal projection of f . As φ increases, the lateral acceleration of the vehicle increases. On the other hand,if the constraint fmax is not sufficiently high, the vehicle could suffer loss of lift. Furthermore, it is interesting to choosean φmax that avoids unexpected flips of the vehicle.

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fxy

f

ϕ

mg

Local Vertical

Reference Trajectory

Local Horizontal

fz

fmax

fmin

CM ZR

YR

XR O

ZB

Figura 3: Analysis of f with respect to its constraints on inclination and magnitude.

4. POLYNOMIAL CHAOS EXPANSION

The analysis of uncertainty propagation and quantification in models has several appliacations in systems engineering.In the presence of stochastic uncertainties, it is important to determine the probabilities of the system properties exceedingspecified critical values or operation limits, and such evaluation can be used to conduct reliability and risk analyses. Inorder to quantify the effects of the disturbance forces Fd on the vehicle trajectory, we employ the nonintrusive formulationof the polynomial chaos expansion in terms of the multivariate Hermite polynomials. The approach considers the analysisof stochastic system responses and uses the PCE as a functional approximation of the mathematical model.

According to Wiener, a stochastic process can be represented through a spectral expansion using orthogonal polyno-mials (Hermite polynomials) as (Sandoval et al., 2012)

g(t, ξ) =

∞∑j=0

gj(t)Ψj(ξ) (14)

where gj(t) is the corresponding deterministic coefficient to be calculated from a limited number of model simulations,ξ = (ξ1, . . . , ξd) the d standard normal independent random variables with zero mean and unit standard deviation and Ψjthe multivariate Hermite polynomials. In practice, the PCE (14) is truncated to a finite number of terms M . This numberis denoted by

M =(p+ d)!

p!d!− 1 (15)

where p is the maximum order of the multivariate polynomial. It is worth noting that the required degree for the polynomialis not know a priori and must be tested to obtain the best accuracy. The multivariate polynomial Ψj(ξ) is given by thetensor product of the corresponding one-dimensional Hermite polynomials Hαjk

Ψj(ξ) =

d∏k=1

Hαjk(ξk) (16)

with αjk the degree of the one-dimensional Hermite polynomials, such that

|αj | =d∑k=1

αjk ≤ p, j = 0, ...,M. (17)

An interesting result of the expansion present in Eq. 14 is that the first two statistical moments of the random variableg are computed from the terms of the PCE, with no need of Monte Carlo sampling methods. Considering the truncateddecomposition, we have:

E[g(t, ξ)] = E

M∑j=0

gj(t)Ψj(ξ)

= g0(t) (18)

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V [g(t, ξ)] = V

M∑j=0

gj(t)Ψj(ξ)

= M∑j=0

g2j (t)V [Ψj(ξ)] (19)

Once the structure of the PCE is obtained for the model output, the deterministic coefficients gj(t) must be calculated.There exist two types of methods for this, the intrusive and the nonintrusive ones. The intrusive method requires to solve asystem of coupled complex equations which could make the implementation of the PCE difficult or sometimes impossible.The nonintrusive approaches use interpolation methods and are very useful when trying to quantify uncertainty in complexstochastic models. The PC coefficients could be obtained through the regression of the mean-square optimization problem.Consider a sample of size N for the variable g(t, ξ) and the parameters ξi with i = 1, . . . , d, at each time instant, thecoefficients ĝ(t) are given by

ĝ(t) = (LTL)−1LTg(t, ξ) (20)

where ĝ(t) is the vector of PC coefficients at instant t and

L =

Φ0(ξ1) Φ1(ξ1) Φ2(ξ1) . . . ΦM (ξ1)Φ0(ξ2) Φ1(ξ2) Φ2(ξ2) . . . ΦM (ξ2)

......

......

...Φ0(ξd) Φ1(ξd) Φ2(ξd) . . . ΦM (ξd)

. (21)We can represent the position components rx(t, ξ), ry(t, ξ), rz(t, ξ) and the control input signals provided by the

MPC controller F̄ c(t, ξ) and φ̄c(t, ξ) by means of the PCE of order M according to Eqs. (14)-(16):

rx(t, ξ) ≈M∑j=0

rx,j (t)Ψj(ξ) (22)

ry(t, ξ) ≈M∑j=0

ry,j (t)Ψj(ξ) (23)

rz(t, ξ) ≈M∑j=0

rz,j (t)Ψj(ξ) (24)

F̄ c(t, ξ) ≈M∑j=0

F̄ c,j (t)Ψj(ξ) (25)

φ̄c(t, ξ) ≈M∑j=0

φ̄c,j (t)Ψj(ξ) (26)

5. COMPUTATIONAL SIMULATIONS

The simulation is implemented in MATLAB/Simulink. The nonlinear 6DOF dynamics of a multirotor aerial vehicle issimulated using a 4th-order Runge-Kutta with a time step of 0.001s. This paper makes use of a more realistic MAV modelthan the one considered in Prado and Santos (2014). An exogenous disturbance Fd is considered during the simulationand it was modelled as follows

Fd =

(0.2 + 0.1ξ1) sin(π + 0.3ξ3)t(0.2 + 0.1ξ2) sin(π + 0.3ξ4)t(0.2 + 0.1ξ3) sin(π + 0.3ξ6)t

(27)where ξi with i = 1, . . . , 6, as aforementioned, are random variables with zero mean and unit standard deviation. Thevehicle is commanded to navigate through the following waypoints ri = [1 1 0]T, w1 = [1 1 1]T, w2 = [2 1 1]T,w3 = [2 2 1]T, w4 = [1 2 1]T, w5 = [1 1 1]T with different combinations of reference speed v and maximum inclinationangle φmax. The polynomial chaos coefficients of the outputs and control signals were calculated considering 300 samples.

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Tabela 2: Polynomial Chaos Results for Different Values of v and φmax.v [m/s] φmax [deg] RMSE(rx) [m] RMSE(ry) [m] RMSE(rz) [m] IADU(φ̄c) IADU(̄f)

10 0.2793 0.3582 0.0705 35.0596 8.26900.5 20 0.2610 0.3300 0.0704 41.9833 8.9152

30 0.2645 0.3322 0.0704 43.2785 7.998510 0.6421 0.6933 0.1242 27.9384 8.4848

1.0 20 0.5557 0.6195 0.1234 48.9406 8.572830 0.5479 0.6076 0.1233 43.6553 8.418510 0.7476 0.6696 0.2167 27.5862 9.7214

2.0 20 0.6777 0.6573 0.2156 75.2982 9.166230 0.6616 0.6512 0.2150 98.7541 9.0742

In order to make a quantitative comparison of the results attained by these combinations, two performace indexeshave been computed. The Root Mean Square Error (RMSE) and the Integral Absolute Derivative Control Signal (IDAU)indexes obtained from the simulations results are presented in Table 2. One can see that as φmax increases, the RMSE(rx)and RMSE(ry) decreases. The same trend is observed in all the reference speeds. This behavior is explained by the factthat a smaller φmax implies in a lower horizontal acceleration (see Fig. 3), which in turn reduces the maneuverabilityof multirotor and consequently the vehicle is not able to reject the disturbance force quickly. As φmax increases, thehorizontal component fxy becomes higher, improving the maneuverability of the vehicle, which in turn provides a fasterdisturbance rejection. However, the RMSE(rz) suffers less influence due to the fact that the vehicle’s rotors have morefreedom to produce vertical thrust. On the other hand, the IADU(φ̄c) had its value increased which shows that theincreasing of φmax produces higher stress of the control actuator, while the values of the IADU(f̄ c) confirms that thealtitude control effort is minimum, even when the vehicle is flying with v = 2 m/s and φmax = 30o.

Now, assigning a specific maximum inclination angle to the vehicle, for example φmax = 30o, and varying thereference speed v, one can see that the standard deviation envelope increases, as illustrated in Fig. 4. The main reason isthe fact that restricting the maneuverability of the vehicle, it reduces the capability of producing horizontal acceleration.

Finally, it is possible to see that even using few samples of the system output, the PCE had a good performance andkept really close to the MC simulation graphs considering 1000 runs (see Fig. 4). Moreover, the computational cost of thePCE simulations considering 1000 runs was much lower when compared with the MC simulations, as illustrated in Table3.

Tabela 3: Computational cost PCE ×MC.PCE [sec] MC [sec]

255 6035

6. CONCLUSIONS

The control method proposed by Prado and Santos (2014) was evaluated by computational simulations consideringthat the vehicle was subjected to a random disturbance force process with amplitude and frequency described by randomvariables with normal distribution. One can conclude that the PCE method is able to propagate the effects of disturbanceforces under the vehicle trajectory. Moreover, a rapid position command yields a larger variability on the system responseand it is necessary to relax the maximum inclination constraint in order to have sufficient lateral control accelerations toovercome the disturbance forces. For future work, we intend to improve the disturbance model in order to consider effectsof Td into the rotational dynamics of the vehicle and analyze parametric uncertainties. Moreover, there is a growinginterest in combining robust control approaches with stochastic control methods in order to be less conservative duringthe control design. Considering this motivation, we intend to design a probabilistic robust control strategies for a MAV.

7. ACKNOWLEDGEMENTS

The authors thank Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacionalde Desenvolvimento Científico e Tecnológico (CNPq) for financial supports.

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9. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.